Wednesday, December 5, 2007

Middle School or SAT Math Activity - The Four Factors Problem

There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.

This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).

The number 12 has 6 positive integer factors: 1,12;2,6;3,4.

(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.

Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?

Eric Jablow said...

Do you think the concept of 'multiplicative function' is accessible to HS students?

Anonymous said...

I wonder if this problem can be expressed as a requirement on the Euler totient (or phi) function. Does not seem obvious since Euler deals with relatively primes.

I do not have a solution.

TC

Oh, we were just working on this sort of problem in my Math Counts class. I drew my problems from an Art of Problem Solving article, but they were too hard---only one student really understood everything. Your problems are better targeted for my kids. Perhaps I will use them as a review when we come back after our Christmas break.

(a) I didn't take the time to do this part, but this is what I think would make a good review for my students.

(b) The two categories will be "prime times prime" and "prime cubed." Right?

(c) The largest 2-digit "prime times prime" is 95 = 5 x 19.

Dave Marain said...

tc, eric--
I would be interested in seeing which states' standards include Number Theory as a 'focal point' throughout all the grades, developed each year with more complexity. I'm sure some form of this exists somewhere but not at the level you're suggesting. Most often, the phi function and its multiplicative nature would only be seen by students engaged in math competition preparation.

denise--
Thank you for those comments. Yes, you nailed the 2 categories.
My experience in working with all grade levels and ability levels makes it slightly easier for me to develop a sequence of questions of increasing difficulty. MathCounts and the Art of Problem Solving challenges have exceptional problems for students to chew on, but they tend to be geared to the higher ability student or require more development for other students. Although I try to leave the more challenging parts of these open-ended questions until the end, this process is never simple for me. I depend on other educators like you to try these out and suggest revisions that would make them better.

mathmom said...

I tried this today with my middle schoolers. Good stuff as always.

I linked to you from my post but I don't see the trackback here, so I guess I'll just post a regular comment...